Deformation Quantization

1. INTRODUCTION Quantum mechanics is often distinguished from classical mechanics by a statement to the effect that the observables in quantum mechanics, unlike those in classical mechanics, do not commute with one another. Yet classical mechanics is meant to give a description (with less precision) of the same physical world as is described by quantum mechanics. One mathematical transcription of this correspondence principle is the that there should be a family of (associative) algebras A ¯ h depending nicely in some sense upon a real parameter ¯ h such that A 0 is the algebra of observables for classical mechanics, while A ¯ h is the algebra of observables for quantum mechanics. Here, ¯ h is the numerical value of Planck's constant when it is expressed in a unit of action characteristic of a class of systems under consideration. (This formulation avoids the paradox that we consider the limit ¯ h → 0 even though Planck's constant is a fixed physical magnitude.) Although the terminology and much of the inspiration comes from physics, noncommu-tative deformations of commutative algebras have also played a role of increasing importance in mathematics itself, especially since the advent of quantum groups about 15 years ago. In the theory of formal deformation quantization, the " family of algebras A ¯ h " is in fact a family ¯ h of associative multiplications on a fixed complex vector space A. More precisely, this family is given by a sequence of bilinear mappings B j : A × A → A for j = 0, 1,. .. so that a ¯ h b =